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Full text of "Mathematical And Physical Papers - Iii"

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the integration being extended over the whole aperture. If it should be necessary to suppose a change of phase to take place in the act of diffraction, such change may be included in the constant B. If, then, / be the intensity,
n2T     /IT .  Zirpx + qy 7   7\2     / ff      27rpx+qy7    7 \2 D2/ =      sin— *   , ™dxdy) + (    cos — ^   ^™ dxdy} ;
\J J         A>         0                     /        \J J           A*          Q                     /
and if / be the total illumination,
-00          ,00
/=          Idpdq.
J _oo J — oo
the limits of a?', y7 being the same as those of #, y.    Hence,
= 1 1 1 /
cos TT~ (p^ — x + qy' — y) dec dy dx dy'.
In the present shape of the integral, we must reserve the integration with respect to p and q till the end; but if we introduce the factor e*0^*^, where the sign — or -f is supposed to be taken according as p or q is positive or negative, we shall evidently arrive at the same result as before, provided we suppose in the end a and /3 to vanish. When this factor is introduced, we may, if we please, integrate with respect to p and q first. We thus get
D2/= limit of
j—- (px — x + qy' — y) dec dy dxf dy' dp dq. Now,
-oo                                                                                                                        /""
eTtt-P cos (kp -Q)dp= cos Q        eTa^ cos kp dp
J -oo                                                                                  J — °°
4- sin Q I     e*"P sin ij) dp
J   -00
f°°             .    .      2a cos 0
= 2cosQ     e-**co*kpdp=    ,,.,-.